Analytic Properties of the Free Energy: the Tricritical Ising Model
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(67) MrlHGQdVCNMrÙ4OQBLCFMr[]d°OQ[SIÒsHRrGJCNGQW CNEHG¼CFBSW [Sh¨lHGQWXOQ\XMPßHMrdH¸8CFEnG BLd4BSRrUpCNMPOØBLRzsH\X[]sGJ\XCNMPGQW [ShÈCFEnG@hj\XGQG@GQdnGQ\N¸LU MPd CFEnGºOJ[]IÏsHRPG ä IÒBS¸]dHGµCFMPO:Ù4GQRrlÑsHRPBSdHG Í ^edÏ[]\NlnGQ\*CF[ºWNqHsns[]\_CCFEHGJMP\*OQRPBSMPI []dnWNGQOQBBLdHl iBLIÒ[]Rr[6lnOfEHM L[mqHWXG>CNEHG ^eWNMPdn¸ ÙHGQRPl CNEHGQ[]\_U 4hj[]\XIKBSRrRrU lHGµÙ4dHGJl(ßVU CFEHGºBSOµCFMr[]d W f Ì]ÍÎÌÄá Z UT j[ T Z T [ T ^edCFEnGKBSßt[Äm]G hj[]\NIwqHR¹B 3\NGQsn\NGQWXGQdVCFWCFEHGÏBSOµCFMP[Sd[Lh¼CFEnGÏWXMPIÏsHRPGJWXC@qHdnMrCfBL\XU IÏMPdnMPIÒBSRIÒ[glHGQR T BSdnl UT BL\NG
(68) CFEnGcsn\NMPIÒBS\_U:Ù4GJRPlHW1[ShgCFEnGCNEHGQ[S\XU CFEHBLC ¼GcOØBLd MrdVCFGQ\XsH\NGµCcBSW*WNsnMPd BSdHlCNEHGQ\XIKBSRn[]sGJ\FBLCN[]\NW EHMrRPG ûBSdHl cBS\XG3OQ[SqHsHRPMrdH¸]W
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(71) CN[þlHGJs4BS\XCNqH\NGJWZhj\N[]ICNEHG°OQ\XMrCNMPOØBLRÈst[]MrdpC8MPd¯CFEHG°IÒBS¸]dnGJCFMrO[S\8MPdÑCFEHGwCNGQIÏsGJ\FBLCNqH\NG lnMP\NGJOJCNMP[]d z\NGQWXsGJOJCNMrm]GJRrU Í `:OJCNq4BSRPRÎU iMÎC<MPW<st[]WXWNMPßnRPGwCN[¯snqnC<CF[]¸SGJCFEnGQ\<CFEHG°ÙH\NWXC@C [ CNGQ\XIÒW:[]dÑCNEHG ZYÑ[LhÔ*ù Í Ì]ÍÎÌÄá BSdHl(WNqHßHW_CFMÎCFqnCNGCFEnGQI MrCNE(CFEHGºhj\NGJGhjGQ\XIÒMr[]dHW BLOJCFMr[]d Ì Z É|ÊfË Ê ËÊ Ê Ë }ÌÍÎËÊ ÊÏÃ[ T W Ì]ÍÎÌø × EnGQ\NG Ì × ¼BSdHl Ê ËÊ BS\XG¯CNEHGàC ¼[)OfEHMP\NBSRkOQ[]IÏst[]dHGQdVCNWK[ShZBíA(B X[]\FBSdHB ÙHGQRPl Í EHMPOfEHGµm]GJ\3hj[]\NI°qnR¹BLCNMP[]dÏ[]dHG:GQIÏsHRP[ÄUgW EHGQd CFEnGZ^eWNMrdH¸wÙ4GJRPlKCNEHGQ[]\_UþMrW¼sHqnC [SdÒB<OµU6RrMPdHlnGQ\ VMÎCFW*\NBSlHMPqnW Ð%BSOµCFWBSWcBSd Mrdnhj\FBS\XGQl\XGQ¸]qHRPBLCF[S\ ]WX[ºCNE4BLC*CFEnGkWNstGQOJCN\NqHI MrW MPdnhj\NBS\NGJl Ù4dHMÎCFG Í ÐU IÏGØBSdHWÏ[Sh@BWNqnMrCfBLßHRPG CF\XqHdHOQBLCFMr[]d[Sh:CFEHG :MrRPßtGQ\_CÒWXs4BSOJG CNEHGZsH\X[]ßHRrGQI [Sh1IÒGQBSWNqn\NMPdn¸wCFEHG8GQdHGJ\N¸SUÒRPGµm]GQRrW3MPW\NGJlHqHOQGJl¯CN[wCFEnGZlHM¹BL¸][]d4BSRrMPÞQBLCFMr[]d [Lh BÑÙ4dHMÎCFGÒlHMPIÏGQdHWXMP[]dHBSR 8BSIÏMPRÎCF[]dnM¹BSd Í ÷ dnGþE4BSWwBSOQOJGQWNW°CN[ÑCNEHGKhj\NGJGKGJdHGQ\X¸SUíßVU W_CFqHl6U6MrdH¸ CFEHG@MPd6hj\FBS\XGQl jCNE4BLC MPW Ð ßtGQE4BØmgMP[]\ [ShÈCFEnGº¸S\N[]qHdnl¯W_CfBLCNG Í c. >c. 1Ï. . ª. ,. $'&Bc. ª. ],. È,. . ç. ß. ß. ç. &. ©. . . ],. Ï. >. -g. å. c=,. c. c. . É. ï. {L|Ñ. c Ù. LyzWm Æ2QÜp. Û. . pÒò. nE GºdnGµägC>OfE4BSsnCNGQ\>¸]MrmSGQW BSdàGµägCN\NGQIÏGQRÎU ßH\XMPGJhÈ\XGJmgMPG b[ShÈCF[]snMPOQWkMrdàOQ[]dnhj[S\NIÒBSRiBSdnl MrdVCFGQ¸S\FBSßHRrGÙ4GQRrl)CFEnGQ[]\_U Í EHGÒ\NGQRrGJmLBSdVCl4BLCfBÑBSßt[]qnCºCFEHG CF\XMPOQ\XMrCNMPOØBLR
(72) ^eWNMrdH¸ÑIÏ[glHGQR BLdHlwMÎCFW1mSBL\NMP[SqHW1MPdVCFGJ¸]\FBSßnRPGcBSdHl°dn[]d6aeMrdVCFGQ¸S\FBSßHRrGlHGJhj[]\XIKBDCFMP[SdHW1BS\NGWNqHIÏIÒBS\NMrÞQGQlºMPd Ú¼E ÍânÍ MPdHMÎCFG
(73) WXMPÞQGG GJOJCFWÈMPd@OJ[]dnhj[]\XIKBSRSBSdHlºMPdVCFGJ¸]\FBLßHRPG
(74) Ù4GJRPl<CFEnGQ[]\_U@BS\NGlHMrWNOJqHWNWXGQl MrdàÚ¼E Í6æ
(75) CFEHMrW¼MrW3B°¸][g[6lÒMPdVCF\X[6lnqHOJCNMP[]dÏCF[°CFEnGZlHGQWXOQ\XMPsnCNMP[]dþ[LhCFEHGZOJ[]IÏsHqnCfBDCFMP[Sd4BSR CNGQOfEHdnMPùpqHG°EHGJ\NGwBSlH[]snCNGQl tß4BSWXGQlo[]d¯CF\XqHdHOØBDCFMP[Sd([Sh*OJ[]dnhj[]\XIKBLRWNs4BSOJG jMPdÑCNEHGhj[SRÎa Rr[ MPdH¸Ò[Sh CFGJdoBSßHßH\XGJmgM¹BDCFGQl ÚkY zCNE4BLC:MPW:GµägCFGJdHWNMÎm]GQRÎUúlHGJWNOJ\NMPßtGQloMrdoÚ¼E ÍiÛ6Í EnG [SqnCFOJ[]IÒGk[LhiIwUw\NGQWXGØBS\XOfEþMPW*sn\NGQWXGQdVCFGJlþMPdÒÚ¼E Ípá EnGQ\NG CFEnG:BLd4BSRrUpCNMPO3W_CF\NqnOJCFqn\NG>[Sh CNEHGkhj\NGJG>GJdHGQ\X¸SU [ShtCFEHGkCN\NMPOJ\NMÎCFMPOQBSRg^eWNMPdn¸IÏ[6lnGQRHMrW*dgqnIÒGJ\NMPOQBSRPRÎUwMrdVm]GQW_CFMr¸VBLCFGJl Í [ MrdVCFGQ\XGQW_CFMPdn¸ßpUgsH\X[6lnqHOJCNW>[LhIU BSd4BLRrUgWNMrWkBS\NGZBSRPWX[\NGQst[]\_CFGQlàMrdKCFEHG<WNBSIÏG<OfE4BSsnCNGQ\ WX[]IÏGKdHMrOQGKsnMPOJCNqH\NGJWw[ShkCNEHGKGµm][]RrqnCFMr[]dí[LhkCFEHGÒWNstGQOJCN\NqHI qHdHlnGQ\°\XGQdH[]\XIKBLRPMPÞQBLCFMr[]d ¸S\N[]qHs 4[ W BSdHl¯CFEHG@dpqHIÏGQ\XMPOØBLRlHGJCNGQ\NIÏMPdHBLCFMr[]dà[Sh
(76) WX[]IÒGºBSIÏsHRrMrCFqnlHGQWkqHWXGJhjqHRhj[]\ lnGJÙ4dHMrdH¸¯qHdHMrmSGQ\XWFBSR*ùpq4BSdVCNMrCFMrGQW Í EnGÒRPBSWXC@OfE4BLsnCFGJ\MrW<lHGJmS[SCFGJl)CF[ÑB WNqHIÏIÒBS\XU([Sh \XGQWXqHRrCNW:BLdHl(BLd¯[SqnCFRr[6[ Ï[]d¯s[SWNWNMrßHRPGZhjq6CFqH\XGºlnGJm]GJRP[]snIÒGJdpCNW Í ,. . O. ! . *. c. -,. O. . c. . c ,. ,. *. U2 -, c. m. >i. ÌSÌ.
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(79) ßVUÏÕ<MPdHWXs4BS\X¸ ÌÄå 4[]\ûCNEHGkßt[6[ \NMÎCNCFGJd ßpUþ^ CFÞJU pWN[SdoBSdHl \N[Sq G ÌÄý Í Ú¼[Sdnhj[]\NIÒBSR1Ù4GQRrlúCFEHGJ[]\XU(MPd(C [àlHMPIÏGQdnWNMP[SdHWZMPW:OfE4BS\NBSOJCNGQ\NMrÞQGJlßVUúBSdMPdnÙ4dnMrCFG dgqnI°ßGJ\@[Sh3OJ[]dHWNGJ\XmLBLCNMP[]d)R¹B W Í Ô*mSGQdMrdoCFEHGÏsH\NGJWNGJdHOQGþ[Lh¼MrdnÙ4dHMÎCFGQRÎUúIÒBSdVUOQ[]dga WNGQ\_m]GJlùpq4BLdpCNMrCNMPGQW tCNEHGCFBSW à[ShWX[]RrmgMrdH¸ÏCNEHGÙHGQRPlÑCFEHGJ[]\XU jCNE4BLC8MPW>OQ[]IÏsHqnCNMPdH¸KBSRPR OQ[]\X\NGQRPBLCFMr[]dÑhjqndHOJCNMP[]dHW MPW8WN[ CF[]qn¸]EúCFE4BDC<MrC8OØBSdoßtGOQ[]IÏsHRrGJCFGJRrU¯OØBL\N\NMrGQlo[]do[SdHRrU hj[]\>CNEH[]WXG°s4BS\_CFMPOJqHR¹BL\NRrUþWNMPIÏsHRrGºOJ[]dnhj[]\XIKBLR1ÙHGQRPlÑCFEHGJ[]\NMrGQW8OQBSRPRrGQlÑIÒMrdHMPIÒBSRiIÏ[6lnGQRPW Y6GJO Ít×6ÍÜ× µÍ `-s4BS\XC8hj\N[]ICFEnGQMP\8IKBLCNEHGQIÒBLCNMPOØBLRMPdVCFGJ\NGQW_C IÏMPdHMrIKBLRiIÒ[glHGJRPWZBL\NG°sHEVUgWNMrOØBSRrRrU dH[SCFG ¼[S\XCFEÒßGJOØBSqnWNG:CNEHGJUÏlHGJWNOJ\NMPßtG CFEHG>WNOQBSRPMrdH¸RrMPIÏMrCû[ShilHMrWNOJ\NGJCNG:W_CfBLCNMPW_CFMPOQBSR4IÏ[6l6a GQRPW:BLC CFEnGQMP\>OQ\XMrCNMPOØBLRs[]MrdVC Í `WNGJOQ[]dHlo[S\NlHGJ\:sHEHBSWNGºCF\FBLdHWNMÎCFMP[Sd¯MrW:MrdHlHGJGQl(OfEHBS\FBSO a CFGQ\XMPÞJGQl°ßVUwCNEHG3lHMÎm]GJ\N¸]GJdHOQGk[Sh4CNEHG3OJ[]\N\XGQR¹BDCFMP[SdRrGQdH¸SCNE 1CFE4BLCûMPWÈCFEHG3\XGØBSWX[]d EVUwCNEHG sHEVU6WXMPOJWw[ShkOJ\NMÎCFMPOQBSRPMÎCeU(\XGQWXqHRrCNWwWNOQBSRPGÏMPdVmLBS\XM¹BSdVC 1[]dnOQGÒCNEHGÏIÒMrOQ\N[SWNOQ[SsHMPOQBSR*WNOQBSRPGÏ[Sh CFEHG@R¹BDCNCFMrOQGZlnGJCfBLMPRPWkMrW GQRPMrIÒMrd4BLCNGQlþßVU CfB gMrdH¸CFEHG@WXOØBSRrMPdH¸ RPMrIÒMÎC Í YgUgWXCFGJIÒW>WNRrMP¸]EVCNRrU [ CFEHGºOQ\XMrCFMrOØBSRst[]MrdpC jWNGQG >Gµh Í × hj[]\>BSdÑGµägCFGJdHWNMÎm]G\XGJmgMPG BS\XGwlHGJWNOJ\NMPßtGQl(ßpU WNqHMÎCfBSßHRrG<lHGJhj[S\NIÒBLCFMr[]dHWk[ShIÒMrdHMPIÒBSRtIÒ[glHGJRPW Í Ô
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